SOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY (x) 5000 ( ) ( )

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1 QUESTION 1 a) Annual Gross Income Less than and less than and less than and less than and less than and less than and above The mean: x = fx f = ( ) 1000 = ( ) The Mode: Mode = L + D 1 i D 1 + D 2 Mid-point (x) 5000 ( ) ( ) Population (f) fx fx Where L = 8000 D 1 = = 50 D 2 = = 90 i = 200 M o = x = The Medium M e = L + f - fbm i 2 Fm Where L = 8000 f = 1000 fbm = 320 fm = 310 i = 2000 :. M e = x =

2 (ii) (b) Since (mean > medium > model), the distribution of annual gross incomes in this suburb is positively skewed (i.e. skewed to the right). The coefficient of variation is given as: CV = Standard deviation x 100% Mean where standard deviation = s = fx fx f f = = ( ) and mean = x = CV = x 100% ( ) = 46.3% ( ) (c) The appropriate curve to show the distribution of taxes is the Lorenz curve, as shown below ( see graph attached) Taxes paid Percentage Cum Taxes Population Percentage Cum Pop QUESTION 2 (a) (i) What quantity of stock to order at any one time. (ii) How frequently to order (iii) Stock holding quantity (iv) Time to place an order (v) Take advantage of discount offers. 2

3 (b) SOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY 2010 Costs involved in stock control are: Order costs Order costs are costs incurred whenever a stock order is generated. There might involve the costs relating to clerical, administrative and managerial activities linked to the order process, costs of transportation, costs of receiving and inspecting orders, costs of finance and accounting support. Purchase cost Purchase cost is the actual cost of purchasing the items from the suppliers. Holding costs Holding costs are those associated with the company holding a fixed quantity of stock over a given period of time. Holding costs can include the cost of capital tied up in the value of the stock, storage costs, (cooling, lighting, security), depreciation, insurance and obsolescence. Stockout costs These are costs incurred when stock is not available. Stockout cost may appear in the form of lost of goodwill or higher prices from another supplier or simply lost profits. (c) Annual Demand : D = 1550 packets Number of working days/year: = 310 days Set-up cost : C s = GH 300 Holding cost : C h = GH 360/packet/year Daily demand : d = 1550 = 5 packets/day 310 Daily production : p = 7750 = 25 packets/day 310 (i) Optimum production lost size is EBQ = 2CsD d C h 1 - p = 2 x 3000 x 1550 (1 5/25) x 360 = = 180 packets (ii) Maximum Inventory = d 1 - p EBQ = (1 5/25) x 180 = 144 packets 3

4 (iii) (iv) (v) Number of production runs = D EBQ Production run time: t 1 = EBQ p t 1 = = 7.2 days Production cycle time: t = EBQ d = = 36 days = = 8.6 times/year = approximately 9 times/year :. Time between production runs is: t 2 = t t1 = = days (vi) Annual inventory cost: TC = DCs + d EBQ Ch EBQ 1 p 2 = 1550 x x 180 x = GH 51,

5 QUESTION 3 SOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY 2010 (a) Let x represents quantity of Doclean (in litres) y represents quantity of Maclean (in litres) Then the problem can be formulated as follows: Max Z = 25x + 18y s.t. 30x + 48y 480 (Labour in stage I) 30x + 75y 600 (Labour in stage II) 5x + 20y 180 (Mix A) 10x + 30y 240 (Mix B) X, y 0 (non-negativity) (b) 30x + 48y = L 1 when x = o, y = 10 (0, 10) y = o, x = 16 (16, 0) 30x + 75y = L 2 when x = o, y = 8 (0, 8) y = o, x = 20 (20, 0) 5x + 20y = L3 when x = o, y = 9 (0, 9) y = o, x = 36 (36, 0) 10x + 30y = L4 when x = o, y = 8 (0, 8) y = o, x = 24 (24, 0) See graph sheet attached for graph. The feasible region is bounded by ABCD. Extreme Point A (0, 9) B (2.667, 8.333) C (6.857, 5.714) D (0, 8) Z = 25x + 18y Hence the optimum production-mix is litres of Doclean and litres of Maclean. 5

6 (c) SOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY 2010 Labour in stage I is a binding constraint Mix B is also a binding constraint Labour in stage II and Mix A are non-binding constraint. (d) (i) 30x + 48y = (1) 10x + 30y = (2) (2) x3 = 30x + 90y = (3) (3) - (1) = 42y = 239 y = :. 10x + 30 ( ) = 240 x= Znew = 25 ( ) + 18 ( ) = = Hence the shadow price for labour in stage I constraint is: = GH 1.36/minute (ii) ie For every 1 minute used after the 480 minutes of the labour in stage I, profit should increase by GH QUESTION 4 (a) Next year s matrices are N = 11, , , M = 11, , , P = M N =

7 (b) SOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY 2010 Let x people buy GH 4.00 denomination and y people buy GH 8.00 denomination (i) For required return of GH (in thousand) x + y = dy = k 1 1 x = y k (ii) We use augmented matrix to obtain the increase of Therefore ¼ ¼ x = 2 - ¼ y -1 ¼ or x = 20,000-56,000 y 4-10, ,000 4 (iii) From. part, x = 20,000-56,000 y 4-10, ,000 4 Thus, 6000 tickets of GH 4.00 denomination and 4,000 tickets of GH 8.00 denomination will be sold. QUESTION 5 a) Let the probability of event A be Pr (A) P (A) denotes the numerical measure of the likelihood of occurrence of event A 0 Pr (A) 1 Pr (A) = 0, the event A is impossible to occur Pr (A) 1, at event a is certain to occur Pr (Ậ) = 1 Pr (A) Pr (A) = 0.5, the event A is just as likely to occur or not. 7

8 b) (i) Blue die 1 1, 1, T 1, 2, T 1, 3, T 1, 4, T 1, 5, T 1, 6, T 1, 1, H 1, 2, H 1, 3, H 1, 4, H 1, 5, H 1, 6, H 2 2, 1, T 2, 2, T 2, 3, T 2, 4, T 2, 1 H 2, 2, H 2, 3,H 2, 4, H (ii) Pr (total score 8, H) = 5 72 (iii) Pr (total score 8) = 10 = (iv) Pr (odd number score less than 7 and a tail) = 6 = c) i. Mutually exclusive events Two or more events which have no common outcomes. If A, B are events that are mutually exclusive, then A B = Ø and Pr (A B) = O Ext events If the sample space S = A U B U C and A, B, C are the only events Independent events Two or more events are independent if the probability of occurrence of one is not influenced by the occurrence or nonoccurrence ie of the other(s). Let M and E represent the event of a choosing a man and an employed person respectively. ii. Pr (M E) = 500 = iii. Pr (E M) = 200 = iv. Pr (M E ) U (M E)) = =

9 QUESTION 6 SOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY 2010 (a) The coefficient of determination can be interpreted as: (i) (ii) a measure of reliability of an estimate the proportion of total variation in the dependent variable as explained by the inclusion of the independent variable(s). (b) (i) The least squares regression equation is given as: Y = a + b x where Y is the profit (in GH 000) X is the sales (in GH 000) a and b are numbers given by: b = n xy - x y n x 2 ( x) 2 a = y - b x n X Y XY X 2 Y :. b = 12 x x = x a = x = Hence, ŷ = x (ii) The regression coefficient is b = ie profits are expected to increase by GH 60.6 for every GH 1000 increase in sales. (iii) (x) when X = 40; Ŷ = (40) =

10 (ß) when X = 400 Ŷ = (400) = (iv) The estimate in (x) is not reliable since x = 40 lies outside the range of values of X used in finding the regression equation. The estimate in (ß) is reliable since X = 400 lies within the range of values of X used in finding the regression equation. (v) The correlation coefficient (r) is: r = n xy - x y [n x 2 ( x) 2 ] [n y 2 ( y) 2] = 12 x x (II) = [12 x ] [12 x ] = :. Coefficient of determination = r 2 x 100% = x 100% = 99% Hence the estimation in b (iii) are 99% reliable. QUESTION 7 (a) The expected monetary value (EMV) of a business decision is the average return that can be expected, taking into account probabilities. The EMV is calculated by multiplying the estimated value of the possible outcomes by the associated probabilities and then summing. The EMV is a useful measure in business as it allows decision-makers to compare alternative decisions. The highest EMV the criterion employed to choose among alternative strategies. 10

11 (b) (i) The Decision Tree (ii) At node a; EMV = x x 0.2 = GH At node c; EMV = x x 0.3 = Gh At node b; EMV = x x 0.5 = GH 8750 At node d; EMV = 0 x x x 0.15 GH 3450 Hence, the best course of action is to expand the business by relocating to a new site. (c) (i) weighing less than 92 kg is. from the standard variable. Z = = From to.. Pr.. weighing less than 9241 = (ii) Standardising, z = = :. Pr weighing more than 97 kg =